Beyond Euler's Method: An Introduction to Runge--Kutta and Implicit Methods

date2026-07-16document_iddoc_bab169344901e0a1be670f7580e163afdescriptionEuler 法の限界を、Runge-Kutta 法による高次精度と陰的方法による安定性の補強へ接続する入口である。prerequisites方向場・Euler 法・誤差と安定性の入口 / 微分法の基本type講義content_typelecturestatusactive

mathdifferential-equationsnumerical-methodslecture

1Introduction

This lecture separates the limitations of Euler's method into accuracy and stability and explains how Runge--Kutta methods and implicit methods address these respective limitations.

2Problem Under Consideration

Consider the initial-value problem

y^{'} = f (t, y), y (t_{0}) = y_{0.}

Euler's method is $y_{n + 1} = y_{n} + h f (t_{n}, y_{n})$ . Because it uses local information only once per step, its accuracy is low. A {stiff problem} is a problem for which accuracy alone would permit a relatively large step size, but the stability of an explicit method requires a very small step size.

It is essential to distinguish accuracy from stability. Accuracy quantifies proximity to the exact solution. Stability determines whether a numerical computation avoids artificially amplifying components that should decay. Even a high-order method fails if its step size violates the applicable stability condition.

3Principle of Runge--Kutta Methods

Runge--Kutta methods evaluate several slopes within a step and combine them to construct the next approximation. The classical fourth-order Runge--Kutta method is

\begin{aligned} k_1&=f(t_n,y_n),\\ k_2&=f(t_n+h/2,y_n+hk_1/2),\\ k_3&=f(t_n+h/2,y_n+hk_2/2),\\ k_4&=f(t_n+h,y_n+hk_3),\\ y_{n+1}&=y_n+\frac{h}{6}(k_1+2k_2+2k_3+k_4). \end{aligned}\begin{aligned} k_1&=f(t_n,y_n),\\ k_2&=f(t_n+h/2,y_n+hk_1/2),\\ k_3&=f(t_n+h/2,y_n+hk_2/2),\\ k_4&=f(t_n+h,y_n+hk_3),\\ y_{n+1}&=y_n+\frac{h}{6}(k_1+2k_2+2k_3+k_4). \end{aligned}

Here, the difference between the exact solution after one step and the result of applying one numerical step from the exact starting value is called the {one-step defect}. If the right-hand side is sufficiently smooth in a neighborhood of the solution and the numerical solution remains in that region, the one-step defect is $O (h^{5})$ . The resulting local Lipschitz continuity with respect to $y$ controls error propagation, and the global error on a fixed finite interval is $O (h^{4})$ . This is the meaning of “fourth order.” Within the asymptotic convergence regime, halving $h$ therefore reduces the global error by approximately a factor of $1 / 16$ .

4Principle of Implicit Methods

Backward Euler is defined by

y_{n + 1} = y_{n} + h f (t_{n + 1}, y_{n + 1}) .

Because the unknown $y_{n + 1}$ occurs on the right-hand side, an equation must be solved at every step. In return, the method has strong stability properties for decay problems such as $y^{'} = a y$ with $a < 0$ .

The use of the slope at the new time level enables the numerical method to respect the direction of a decaying solution more reliably. The associated cost is the need to establish existence and uniqueness of a solution to the algebraic equation at each step and, in the nonlinear case, to update an approximation iteratively. Newton's method is a standard iteration that uses a derivative to form a local linearization. Uniqueness of the root alone, however, does not imply convergence of Newton iteration. A practical computation must monitor a scaled residual, the update magnitude, and an iteration limit selected for the problem.

5Example

For $y^{'} = - 10 y, y (0) = 1$ , Forward Euler gives $y_{n + 1} = (1 - 10 h) y_{n}$ . With $h = 0.3$ , its amplification factor is $- 2$ , so the method is unstable. Backward Euler instead gives

y_{n + 1} = y_{n} - 10 h y_{n + 1},

and hence

y_{n + 1} = \frac{1}{1 + 10 h} y_{n} .

The numerical solution therefore decays for every $h > 0$ .

In this example, the exact solution $e^{- 10 t}$ decreases monotonically. When Forward Euler uses $h = 0.3$ , multiplication by $- 2$ reverses the sign and increases the magnitude at every step. This behavior is an artifact of the numerical method, not a property of the differential equation.

6Selection Criteria

Apply a numerical method to the test equation $y^{'} = λ y$ with $λ \in C$ . Denote its one-step {amplification factor} by $R (z)$ , where $z = h λ$ , and let $Re z$ denote the real part of $z$ . A method is {A-stable} if $| R (z) | [PARSE ERROR: Undefined("Command(\"le\")")] 1$ throughout the left half-plane $Re z < 0$ . An A-stable method is {L-stable} if it additionally satisfies $R (z) \to 0$ as $| z | \to \infty$ within the left half-plane. Backward Euler has both properties.

Situation	Candidate	Rationale
Smooth, nonstiff problem	Classical fourth-order Runge--Kutta method	Provides high-order accuracy with modest implementation complexity
Stiff problem containing both rapidly decaying and slow components	An implicit method with A-stability or L-stability	Handles decay modes along the negative real axis stably
Computation requiring an empirical accuracy check	Comparison after halving the step size	Assesses error reduction within the asymptotic convergence regime

7Scope and Limitations

Runge--Kutta methods are standard instruments for improving accuracy, but a stiff problem may require an extremely small step size when an explicit Runge--Kutta method is used. Not every implicit method is unconditionally stable; the stability region must be examined for each method. A step-halving comparison by itself is not a rigorous bound on the exact error.

8Exercises

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